I've plotted several locations based in Auckland, New Zealand, using Grid NZGD2000:ESPG:2193. When I load QuickMapServices and set to the same grid, The basemap produced is situated off Omen. 
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1I doubt the problem comes from QMS. I would rather think your data is misplaced. Check your data source projection. If possible, publish some examples of X/Y values.– gisnsideCommented Jun 18, 2018 at 7:53
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How do you "set to the same grid"? If the basemap is in EPSG:3857, you better leave it in tthat CRS. Do the layers align if you set the project CRS to EPSG:3857?– AndreJCommented Jun 18, 2018 at 11:09
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I just change it in properties. When left as default, no basemap shows up. X/Y shown below: 5920705 1755253 5918819 1753290 5932578 1753626 5976580 1755875 5912447 1763803– RF1Commented Jun 18, 2018 at 23:07
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2 Answers
Per the comments, you're doing it wrong.
You can't simply change a layer's CRS if it was defined as another.
How to resolve: Make sure your layer in the original EPSG:2193 CRS.
If after you load the basemap, the canvas doesn't change to EPSG:3857, change the canvas to EPSG:3857, and zoom to your layer's location in New Zealand.
You mentioned in the comments that updating to the new version of QGIS solved the problem. I assume you updated from a 2.x version to a 3.x version.
The solution to your original problem would have been to turn on "on-the-fly reprojection." This feature allows multiple layers in different CRS to be displayed as if they were all in the same CRS without creating a new version of each layer.
In the QGIS 2.x series, on-the-fly reprojection was an option that could be toggled in the project properties. In QGIS 3.0 and later, on-the-fly reprojection is always on. That's why updating to the new version of QGIS solved your issue.
